Maximum Likelihood Spectral Fitting: The Batchelor Spectrum
Bibliographic record
Abstract
A simple technique for fitting spectra that is applicable to any problem of adjusting a theoretical spectral form to fit observations is described. All one needs is a functional form for the theoretical spectrum and an estimate for the instrumental noise spectrum. The method, based on direct application of the maximum likelihood approach, has several advantages over other fitting techniques. 1) It is unbiased in comparison with other least squares or cost function–based approaches. 2) It is insensitive to dips and wiggles in the spectrum, due to the small number of fitted parameters. It is also robust because the range of wavenumbers used in the fit is held fixed, and the built-in noise model forces the routine to ignore the spectrum as it gets down toward the noise level. 3) The method provides a theoretical estimate for error bars on the fitted Batchelor wavenumber, based on how broad or narrow the likelihood function is in the vicinity of its peak. 4) Statistical quantities that indicate how well the observed spectrum fits the theoretical form are calculated. This is extremely useful in automating analysis software, to get the computer to automatically flag “bad” fits. The method is demonstrated using data from the Self-Contained Autonomous Microstructure Profiler (SCAMP), a free-falling temperature microstructure profiler. Maximum likelihood fits to the Batchelor spectrum are compared to the SCAMP-generated fits and other least squares techniques, and also tested against pseudodata generated by Monte Carlo techniques. Pseudocode outlines for the spectral fit routines are given.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.012 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.007 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".